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Calculus, MET1 2024 VCAA 3

Let  \(g: R \backslash\{-3\} \rightarrow R, \ g(x)=\dfrac{1}{(x+3)^2}-2\).

  1. On the axes below, sketch the graph of  \(y=g(x)\),  labelling all asymptotes with their equations and axis intercepts with their coordinates.   (3 marks)


  2. Determine the area of the region bounded by the line  \(x=-2\),  the \(x\)-axis, the \(y\)-axis and the graph of \(y=g(x)\).   (2 marks)

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a.

b.    \(\dfrac{10}{3}\ \text{sq units}\)

Show Worked Solution

a.    \(y\text{-intercept:}\ x=0\)

\(y=\dfrac{1}{(0+3)^2}-2=-\dfrac{17}{9}\)

\(x\text{-intercepts:}\ y=0\)

\(\dfrac{1}{(x+3)^2}-2\) \(=0\)
\((x+3)^2\) \(=\dfrac{1}{2}\)
\(x+3\) \(=\pm\dfrac{1}{\sqrt{2}}\)
\(x\) \(=-3\pm\dfrac{1}{\sqrt{2}}\)

b.   \(\text{Area is below}\ x\text{-axis:}\)

  \(\text{Area}\) \(=-\displaystyle\int_{-2}^0 (x+3)^{-3}-2\,dx\)
    \(=-\left[\dfrac{1}{-1}(x+3)^{-1}-2x\right]_{-2}^0\)
    \(=-\left[\dfrac{-1}{x+3}-2x\right]_{-2}^0\)
    \(=-\left[\dfrac{-1}{3}-\left(\dfrac{-1}{-2+3}-2(-2)\right)\right]\)
    \(=-\Big[\dfrac{-1}{3}-(-1+4)\Big] \)
    \(=\dfrac{10}{3}\ \text{u}^{2}\)
♦ Mean mark (b) 40%.

Graphs, MET1 EQ-Bank 1

Let  \(\displaystyle f:[-3,-2) \cup(-2, \infty) \rightarrow R, f(x)=1+\frac{1}{x+2}\).

  1. On the axes below, sketch the graph of \(f\). Label any asymptotes with their equations, and endpoints and axial intercepts with their coordinates.   (3 marks)

  1. Find the values of \(x\) for which \(f(x) \leq 2\).   (2 marks)

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a.
   

b.   \(x\in [-3, -2)\cap (-1, \infty)\)

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a.
     

b.    \(\text{From graph }f(x)\leq -2\ \text{ for}\ -3\leq x <2\ \text{ and when }\ x\geq -1\) 

\(\rightarrow x\in [-3, -2)\cap (-1, \infty)\)

Graphs, MET2 2020 VCAA 5 MC

The graph of the function  `f:D rarr R,f(x)=(3x+2)/(5-x)`, where `D` is the maximal domain, has asymptotes

  1. `x=-5,y=(3)/(2)`
  2. `x=-3,y=5`
  3. `x=(2)/(3),y=-3`
  4. `x=5,y=3`
  5. `x=5,y=-3`
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`E`

Show Worked Solution
`f(x)` `=(3x+2)/(5-x)`  
  `=(-(15-3x)+17)/(5-x)`  
  `=-3+17/(5-x)`  

 
`text(Vertical asymptote:)\ \ x=5`

`text(Horizontal asymptote:)\ \ y=-3`

`=>E`

Graphs, MET1 2021 VCAA 4

  1. Sketch the graph of  `y = 1-2/(x-2)`  on the axes below. Label asymptotes with their equations and axis intercepts with their coordinates.   (3 marks)

     

     

  2. Find the values of  `x`  for which  `1-2/(x-2) >= 3`.   (1 mark)

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a. 

b. `x in [1, 2)`

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a.   `text(Asymptotes:)`

`x=2`

`text(As)\ \ x→ +-oo, \ y→1\ \ =>\ text(Asymptote at)\ \ y=1`

`ytext(-intercept at)\ (0,2)`

`xtext(-intercept at)\ (4,0)`

♦ Mean mark part (b) 32%.

b.   `text(By inspection of the graph:)`

`1-2/(x-2) >=3\ \ text(for)\ \ x in [1, 2)`

Graphs, MET2-NHT 2019 VCAA 4 MC

The graph of the function  `ƒ : D → R, \ f(x) = (2x -3)/(4 + x)`, where `D` is the maximal domain, has asymptotes

  1.  `x = –4, \ y = 2`
  2.  `x = (3)/(2), \ y = –4`
  3.  `x = –4, \ y = (3)/(2)`
  4.  `x = (3)/(2), \ y = 2`
  5.  `x = 2, \ y = 1`
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`A`

Show Worked Solution
`f(x)` `= (2x + 8 – 11)/(x +4)`
  `= 2 – (11)/(x + 4)`

 
`:. \ text(Asymptotes at) \ \ x = –4, y = 2 `